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CRunDec: a C++ package for running and decoupling of the strong coupling and quark masses

Barbara Schmidt, Matthias Steinhauser

arXiv:1201.6149v1hep-ph

TL;DR

Scale- and flavour-dependent QCD parameters require running and decoupling routines, while heavy-quark masses also need transformations among renormalization schemes. CRunDec implements these calculations in C++, extending the Mathematica package RunDec; its examples include a top-quark result of mt(mt) = 164.0 GeV.

  • Problem

    The strong coupling and light-quark masses depend on scale and active flavours, requiring running and decoupling formulae, while heavy-quark masses may need conversion between renormalization schemes.

    Source: Program summary · p. 2

  • Method

    CRunDec implements running, decoupling, and mass-conversion routines in C++, including a new routine that simultaneously solves for αs and mq.

    Source: §1 · p. 3, §2.2 · p. 5, §2.3 · p. 6

  • Results

    The top-quark example computes mt(mt) = 164.0 GeV from the on-shell mass.

    Source: Compute mt(mt) from the top quark… · p. 9

  • Takeaways & Limitations

    CRunDec brings RunDec routines into C++ with additions and improvements for running, decoupling, and quark-mass conversions.

    Source: Program summary · p. 2, §1 · p. 3, §2.3 · p. 6

    CRunDec includes decoupling relations only for on-shell heavy quarks and omits automatic AsRunDec flavour selection, recommending explicit AlL2AlH and AlH2AlL routines instead.

    Source: §2.4 · p. 6, §2.5 · p. 7

Abstract

from arXiv · show

In this paper we present the {\tt C++} package {\tt CRunDec} which implements all relevant formulae needed for the running and decoupling for the strong coupling constant and light quark masses. Furthermore, several formulae are implemented which can be used to transform the heavy quark masses among different renormalization schemes. {\tt CRunDec} is the {\tt C++} version on the {\tt Mathematica} package {\tt RunDec} containing several updates and improvements.

Program summary

The strong coupling and light-quark masses depend on energy scale and active-flavour number. CRunDec collects formulae and procedures for their running and decoupling, with an adaptive Runge–Kutta procedure for differential equations.

  • The strong coupling and light-quark masses depend on the energy scale and, for the coupling, the number of active quark flavours.The program provides procedures for evaluating these quantities using the collected formulae.Source: Program summary · p. 2
  • CRunDec collects relevant formulae and procedures for running and decoupling the strong coupling and light-quark masses using state-of-the-art correction terms.Source: Program summary · p. 2
  • The package is implemented in C++, and its differential equations are solved with an adaptive Runge–Kutta procedure.Source: Program summary · p. 2
  • Results may be nonsensical when input parameters are chosen unphysically.Source: Program summary · p. 2

1. Introduction

Scale dependence and flavour-threshold decoupling motivate routines for evolving QCD parameters. CRunDec brings RunDec routines into C++ with additions and improvements, and offers usage examples.

1. Introduction

  • Running requires scale evolution of the strong coupling and quark masses, with decoupling relations applied when crossing a flavour threshold.Source: §1 · p. 2
  • CRunDec converts important RunDec routines to C++, adds new routines, and improves others for use outside a computer algebra system.Its public class components contain functions known from the Mathematica package RunDec [1].Source: §1 · p. 3
  • The paper presents typical usage examples, and complete programs are supplied in example.cc.Source: §1 · p. 3

2. Structure of CRunDec

CRunDec exposes running, mass-conversion, and decoupling routines through a C++ class. Its functions include new running support and modified mass-conversion implementations, while combined running and decoupling is called explicitly rather than automatically.

2. Structure of CRunDec

  • The class exposes functions corresponding to RunDec [1], with additions, and overloads several functions so the active-flavour argument can be omitted when set on the object.GetNf() retrieves that setting and SetNf(int nf) changes it.Source: §2 · pp. 3–4

2.1. Input parameters

  • Frequently used input parameters are predefined and can be redefined for calculations using other numerical values.Source: §2.1 · p. 4

2.2. Functions related to the running of αs and mq

  • AsmMSrunexact is new and simultaneously solves the differential equations for αs and mq from initial values at µ0, returning both at µ.The returned struct has components for αs(µ) and mq(µ).Source: §2.2 · p. 5

2.3. Functions relating different mass definitions

  • The mass-conversion functions cover multiple mass definitions, and mq[] supplies light-quark-mass effects when populated before a call.Its elements are zero by default.Source: §2.3 · pp. 5–6
  • mMS2mRGImod is new and evaluates c(x) using x = 2β0αs/π rather than x = αs/π.The mMS2mSI implementation also avoids an intermediate ΛQCD computation and is described as more perturbatively stable, particularly at lower renormalization scales.Source: §2.3 · p. 6

2.4. Functions related to the decoupling of heavy quarks

  • CRunDec includes decoupling relations only for on-shell heavy quarks, and its coupling-decoupling functions include four-loop relations [6] [7].The nf argument in these functions refers to the effective theory's number of flavours.Source: §2.4 · pp. 6–7

2.5. Functions related to the combination of running and decoupling

  • Combined running and decoupling uses explicit AlL2AlH and AlH2AlL routines instead of an automatic AsRunDec function.The paper cites inconvenience of always decoupling at µth = Mth and ambiguities in choosing active flavours as reasons for this choice.Source: §2.5 · p. 7

3. Typical examples

The paper introduces typical CRunDec usage examples and points readers to the accompanying source file for complete programs.

3. Typical examples

  • The examples show typical uses of CRunDec, with only the package-related code displayed in the paper.Source: §3 · p. 7
  • Complete example programs are provided in the accompanying example.cc file.Source: §3 · p. 7

Running of αs from Mt to Mb with five active flavours

The examples demonstrate running αs across flavour thresholds and converting quark masses between renormalization schemes. They report results for αs at MZ and Mb, and for bottom and top scale-invariant masses and on-shell charm and bottom masses.

Running of αs from Mt to Mb with five active flavours

Compute mb(mb) from mb(10 GeV)

Compute mt(mt) from the top quark on-shell mass

Compute on-shell charm and bottom quark mass from mc(mc) and mb(mb)

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